Orbits
Satellites
- Planets and Satellites are kept in orbit by gravitational forces
- We can find the time taken for a satellite to make one orbit by rearranging equations.
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\[\frac{mv^2}{r}=\frac{GMm}{r^2}\]
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\[v^2=\frac{GM}{r}\]
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\[v=\sqrt{ \frac{GM}{r} }\]
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\[v=\sqrt{ \frac{GM}{r} }\]
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\[\frac{2\pi r}{T}=\sqrt{ \frac{GM}{r} }\]
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\[2\pi r=\sqrt{ \frac{GM}{r} }T\]
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\[T=\frac{2\pi r}{\sqrt{ \frac{GM}{r} }}\]
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\[T=\frac{2\pi^2r^2}{\frac{GM}{r}}\]
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\[T^2=\frac{4\pi^2r^3}{GM}\]
- Kepler's third law: \(\(T^2=\left( \frac{4\pi^2}{GM} \right)r^3\)\)
Worked example
- Mean distance of \(4.5\times10^{12}m\) from the sun
- Mass of the sun is \(1.99\times 10^{30}\)
- \(T^2=\left( \frac{4\pi^2}{(6.67\times 10^{-11})\times(1.99\times 10^{30})} \right)\times(4.5\times 10^{12})^3\)
- \(T=5.206\times 10^9\) seconds or \(165\) earth years.
Worked Example 2
- Radius of earth = \(6.370\times10^6\)
- T= 24 hours = \(86400\) seconds
- Mass of earth = \(5.97\times10^{24}\)
- Use \(T^2=\left( \frac{4\pi^2}{GM} \right)r^3\)
- Rearrange to find \(r\), then subtract the radius of the earth.
Escape Velocity
\[E_{k}=E_{GV}$$
$$\frac{1}{2}mv^2=\frac{GMm}{r}$$
$$mv^2=\frac{2GMm}{r}$$
$$v^2=\frac{2GM}{r}$$
$$v=\sqrt{ \frac{2GM}{r}}\]
Deriving the equation for the angular motion of an orbiting satellite
- \(v=\sqrt{ \frac{GM}{r} }\)
- \(v=\omega r\)
- \(\omega r=\sqrt{ \frac{GM}{r} }\)
- \(\omega=\sqrt{ \frac{GM}{r^3} }\)
Questions
- a) Gravitational field strength is a scalar quantity.
b) - a) 9.81 N/kg
b) 19.62 N
c) -58.86
d)kinetic
e) gravitational potential - 180J
- a) -0.0667
b) -0.3335
c) \(600\times -0.03335=-20.01\)
d) outwards - a) 1.08
b) 42226910.18m - a) 9.81344 N/kg
b) 8.1102 N/kg